The formulae given by equations (REF )-(REF ) are a recursive formula of the Nth-order generalized DT for the Manakov system (REF ) and (REF ). Although it is possible to give the \((3N)\times (3N)\)  determinant representation by using the so-called crum theorem [1]}. But we prefer to use a recursive formula, because it is easy to construct higher-order rogue wave solutions with the computer. Some interesting higher-order rogue wave solutions are obtained for the Manakov system in the following section.
